complex analysis - k. houston

Visualising complex functions In Real Analysis we could draw the graph of a function.. Wehave an axis for the variable and an axis for the value, and so we can draw the graph of the func

func-of convergence for these complex series.

We start by defining domains in the complex plane Thisrequires the prelimary definition

Definition 1.1

The ε-neighbourhood of a complex number z is the set of

com-plex numbers {w ∈ C : |z − w| < ε} where ε is positive number.Thus the ε-neigbourhood of a point z is just the set of pointslying within the circle of radius ε centred at z Note that itdoesn’t contain the circle

Definition 1.2

A domain is a non-empty subset D of C such that for every

point in D there exists a ε-neighbourhood contained in D

Examples 1.3

The following are domains

(i) D = C (Take c ∈ C Then, any ε > 0 will do for an neighbourhood of c.)

ε-(ii) D = C\{0} (Take c ∈ D and let ε = 1

2(|c|) This gives aε-neighbourhood of c in D.)

(iii) D = {z : |z − a| < R} for some R > 0 (Take c ∈ C and let

ε = 12(R − |c − a|) This gives a ε-neighbourhood of c in D.)

Example 1.4

The set of real numbers R is not a domain Consider anyreal number, then any ε-neighbourhood must contain somecomplex numbers, i.e the ε-neighbourhood does not lie in thereal numbers

We can now define the basic object of study

Definition 1.5

Let D be a domain in C A complex function, denoted f : D →

C, is a map which assigns to each z in D an element of C, thisvalue is denoted f (z)

Common Error 1.6

Note that f is the function and f (z) is the value of the function

at z It is wrong to say f (z) is a function, but sometimes peopledo

Examples 1.7

(i) Let f (z) = z2 for all z ∈ C

(ii) Let f (z) = |z| for all z ∈ C Note that here we have acomplex function for which every value is real

(iii) Let f (z) = 3z4 − (5 − 2i)z2+ z − 7 for all z ∈ C All complexpolynomials give complex functions

(iv) Let f (z) = 1/z for all z ∈ C\{0} This function cannot beextended to all of C

Remark 1.8

Functions such as sin x for x real are not complex functionssince the real line in C is not a domain Later we see how toextend the concept of the sine so that it is complex function

on the whole of the complex plane

Obviously, if f and g are complex functions, then f + g,

f − g, and f g are functions given by (f + g)(z) = f (z) + g(z),(f − g)(z) = f (z) − g(z), and (f g)(z) = f (z)g(z), respectively

We can also define (f /g)(z) = f (z)/g(z) provided that g(z) 6= 0

on D Thus we can build up lots of new functions by theseelementary operations

The aim of complex analysis

We wish to study complex functions Can we define ation? Can we integrate? Which theorems from Real Analysiscan be extended to complex analysis? For example, is there aversion of the mean value theorem? Complex analysis is es-sentially the attempt to answer these questions The theorywill be built upon real analysis but in many ways it is easierthan real analysis For example if a complex function is dif-ferentiable (defined later), then its derivative is also differen-tiable This is not true for real functions (Do you know an ex-ample of a differentiable real function with non-differentiablederivative?)

differenti-Real and imaginary parts of functions

We will often use z to denote a complex number and we willhave z = x + iy where x and y are both real The value f (z)

is a complex number and so has a real and imaginary part

We often use u to denote the real part and v to denote theimaginary part Note that u and v are functions of z

We often write f (x + iy) = u(x, y) + iv(x, y) Note that u is

a function of two real variables, x and y I.e u : R2 → R.Similarly for v

Visualising complex functions

In Real Analysis we could draw the graph of a function Wehave an axis for the variable and an axis for the value, and so

we can draw the graph of the function on a piece of paper.For complex functions we have a complex variable (that’stwo real variables) and the value (another two real variables),

so if we want to draw a graph we will need 2 + 2 = 4 realvariables, i.e we will have to work in 4-dimensional space.Now obviously this is a bit tricky because we are used to 3space dimensions and find visualising 4 dimensional spacevery hard

Thus, it is very difficult to visualise complex functions ever, there are some methods available:

How-(i) We can draw two complex planes, one for the domain andone for the range

(ii) The two-variable functions u and v can be visualised arately The graph of a function of two variables is a sur-face in three space.

sep-u(x, y) = cos x + sin y and v(x, y) = x2− y2

(iii) Make one of the variables time and view the graph assomething that evolves over time This is not very helpful

Defining ez, cos z and sin z

First we will try and define some elementary complex tions to play with How shall we define functions such as ez,cos z and sin z? We require that their definition should coincidewith the real version when z is a real number, and we wouldlike them to have properties similar to the real versions of thefunctions, e.g sin2z + cos2z = 1 would be nice However, sineand cosine are defined using trigonometry and so are hard togeneralise: for example, what does it mean for a triangle tohave an hypotenuse of length 2 + 3i? The exponential is de-fined using differential calculus and we have not yet defineddifferentiation of complex functions

func-However, we know from Real Analysis that the functionscan be described using a power series, e.g.,

sin x = x − x

3

3! +

x55! − · · · =

2n

To answer this we will have to study complex series and

as the theory of real series was built on the theory of realsequences we had better start with complex sequences

Complex Sequences

The definition of convergence of a complex sequence is thesame as that for convergence of a real sequence

Definition 1.11

A complex sequence hcni converges to c ∈ C, if given any ε > 0,

then there exists N such that |cn− c| < ε for all n ≥ N

 4 − 3i7

n

=

4 − 3i7

= |z|

n + 1

→ 0 as n → ∞

The last part is true because for fixed z the real number |z| is

of course a finite constant

So by the ratio test P∞

n=0

z n

n! converges for all z ∈ C

The following is an example with some of the small detailmissing This is how I would expect the solution to be given if

I had set this as an exercise

(−1)n z2n+1(2n + 1)!

Then

an+1

an

=

... data-page="7">

Complex Series

Now that we have defined convergence of complex sequences

we can define convergence of complex series

Definition 1.19

A complex. .. the paradigm1 wewill be using We can apply results from real analysis to pro-duce results in complex analysis In this case we take themodulus, but we can also take real and imaginary...

In real analysis we have some great ways to tell if a series

is convergent, for example, the ratio test and the integral test.Can we use the real analysis tests in complex analysis?